English

The Cauchy-Schlomilch transformation

Classical Analysis and ODEs 2010-04-15 v1

Abstract

The Cauchy-Schl\"omilch transformation states that for a function ff and a,b>0a, \, b > 0, the integral of f(x2)f(x^{2}) and af((axbx1)2af((ax-bx^{-1})^{2} over the interval [0,)[0, \infty) are the same. This elementary result is used to evaluate many non-elementary definite integrals, most of which cannot be obtained by symbolic packages. Applications to probability distributions is also given.

Keywords

Cite

@article{arxiv.1004.2445,
  title  = {The Cauchy-Schlomilch transformation},
  author = {T. Amdeberhan and M. L. Glasser and M. C. Jones and V. H. Moll and R. Posey and D. Varela},
  journal= {arXiv preprint arXiv:1004.2445},
  year   = {2010}
}
R2 v1 2026-06-21T15:10:23.608Z